Title: Character degree graph and Huppert’s ρ-σ conjecture
Character Theory is one of the strong tools in the theory of finite groups, and, given a finite group G, the study of the set cd(G)={θ(1)|θϵ Irr(G)}, of all degrees of the irreducible complex characters of G, has an important role in finite group theory. Associating a graph to the degree-set is one of the method to approach this set. The character degree graph Δ(G) is defined as the graph whose vertex set is the set of all the prime numbers that divide some θ(1)∈ cd(G), while a pair (p,q) of distinct vertices p and q belongs to the edge set if and only if pq divides an element in cd(G). So far, many studies have been done on this graph. In this talk, we will discuss the recent development obtained on this graph and finally focus on a new result on Huppert’s ρ-σ conjecture, which is derived from the recent development on this graph.
Title: Symmetric groups and the Heisenberg category
We will see how induction and restriction give an action of the Heisenberg algebra on the category of representations of symmetric groups. We will discuss how this inspired Khovanov’s definition of the Heisenberg category, as well as some recent developments.
Title: Problems on characters involving two primes
In the first part of the talk we will see that if G is a nontrivial finite group, then for every pair of primes {p,q} there is some nontrivial irreducible character of G whose degree is not divisible by p nor q. This result will allow us to characterize groups in which all irreducible characters of degree not divisible by p nor q are linear. In the second part of my talk, I will discuss on what can be said about the field of values of such a character of {p, q}'-degree. This talk is based in joint works with E Giannelli, N. Hung and M. Schaeffer Fry.
Title: Picard groups for blocks with normal defect groups
Let G be a finite group, B a p-block of OG, O a complete DVR. The Picard group of B is the group of auto-Morita equivalences of B and it revealed itself to be a useful tool, for example for dealing with Donovan conjecture. However, it is also interesting in its own right, since it has the structure of a finite group, when O has char 0.
In this talk we will give an introduction to Picard groups for blocks and then present joint work with Livesey on blocks with normal defect groups, providing evidence to a conjecture on basic Morita equivalences.
Title: Groups in which the centralizer of any non-central primary element is maximal.
In this talk, we investigate the structure of a finite group G whose centralizer of each primary element is maximal in G. This is a question raised by Zhao, Chen and Guo in "Zhao, Xianhe; Chen, Ruifang; Guo, Xiuyun Groups in which the centralizer of any non-central element is maximal. J. Group Theory 23 (2020), no. 5, 871–878".
In this talk, we also provide an independent result focused on the centralizers of primary elements in finite simple groups.
Title: On the number of p-elements in finite groups.
Given a finite group G and a prime p dividing its order, we consider the ratio between the number of p-elements and the order of a Sylow p-subgroup of G. Frobenius proved that this ratio is always an integer, but no combinatorial interpretation of this number seems to be known.
We will talk about the search for a lower bound on this ratio in terms of the number of Sylow p-subgroups of G.
Title: Character Triple Conjecture for Groups of Lie Type.
Dade’s Conjecture is an important conjecture in representation theory of finite groups. It implies most of the, so called, global-local conjectures. In 2017, Späth introduced a strengthening of Dade’s Conjecture, called the Character Triple Conjecture, which describes the Clifford theory of corresponding characters. Moreover, she proved a reduction theorem, namely that if her conjecture holds for every quasisimple group, then Dade’s Conjecture holds for every finite group. Extending ideas of Broué, Fong and Srinivasan we provide a strategy to prove the Character Triple Conjecture for quasisimple groups of Lie type in the nondefining characteristic.
Title: The McKay-Navarro Conjecture: the Conjecture That Keeps on Giving!
The McKay conjecture is one of the main open conjectures in the realm of the local-global philosophy in character theory. It posits a bijection between the set of irreducible characters of a group with p’-degree and the corresponding set in the normalizer of a Sylow p-subgroup. In this talk, I’ll give an overview of a refinement of the McKay conjecture due to Gabriel Navarro, which brings the action of Galois automorphisms into the picture. A lot of recent work has been done on this conjecture, but possibly even more interesting is the amount of information it yields about the character table of a finite group. I’ll discuss some recent results on the McKay—Navarro conjecture, as well as some of the implications the conjecture has had for other interesting character-theoretic problems.
Title: The generalization of a theorem on real valued characters
The Theorem of Ito-Michler, one of the most celebrated results in character theory of finite groups, states that a group has a normal abelian Sylow p-subgroup if and only if the prime number p does not divide the degree of any irreducible character of the group.
Among the many variants of the theorem, there exists one, due to Dolfi, Navarro and Tiep, which involves only the real valued irreducible characters of the group, and the prime number p = 2.
This variant, however, fails if we consider a prime number different from 2, and any generalization in this direction seems hard, due to some specific properties of real valued characters.
This talk proposes a new way to approach the problem, which takes into account a different subset of the irreducible characters, however related with real valued characters. This new approach has already been partially successful and it may suggest a way to generalize also other similar results.
Title: Degrees of characters in the principal block
Let G be a finite group and let p be a prime. The set of complex irreducible characters in the principal p-block of G is rich enough that their degrees encode information of the structure of the group G. We study the case where the set of degrees of characters in the principal p-block of G has size at most 2, finding information about the structure of G and its Sylow p-subgroups. We will also show some related results on similar problems for arbitrary p-blocks.
Title: Bounding p-regular conjugacy classes and p-Brauer characters in finite groups.
We discuss two closely related problems on bounding the number of p-regular conjugacy classes of a finite group G and bounding the number of irreducible p-Brauer characters of G or a block of G. Among other results we will show that the number of p-regular classes of a finite group G is bounded below by 2√(p−1)+1−kp(G), where kp (G) is the number of classes of p-elements of G. This and the celebrated Alperin weight conjecture imply the same bound for the number of irreducible p-Brauer characters in the principal p-block of G. We also discuss the bounds in the minimal situation when G has a unique class of nontrivial p-elements, which have applications to the study of principal blocks with few characters. The talk is based on joint works with A. Moretó, with A. Maroti, and with B. Sambale and P.H. Tiep.
Title: Fields of definition for linear representations
A classical theorem of Richard Brauer asserts that every finite-dimensional non-modular representation ρ of a finite group G defined over a field K, whose character takes values in k, descends to k, provided that k has suitable roots of unity. If k does not contain these roots of unity, it is natural to ask how far ρ is from being definable over k. The classical answer to this question is given by the Schur index of ρ, which is the smallest degree of a finite field extension l/k such that ρ can be defined over l. In this talk, based on joint work with Nikita Karpenko, Julia Pevtsova and Dave Benson, I will discuss another invariant, the essential dimension of ρ, which measures ”how far” ρ is from being definable over k in a different way by using transcendental, rather than algebraic field extensions. I will also talk about recent results of Federico Scavia on essential dimension of representations of algebras.
Title: p-constant characters of finite groups
Let p be a prime number; an irreducible character of a finite group G is called p-constant if it takes a constant value on all the elements of G whose order is divisible by p (p-singular elements). Irreducible characters of p-defect zero are, by a classical result or R. Brauer, an important instance of this class of characters: they take value zero on every p-singular element. I will present some results on faithful p-constant characters of 'positive defect'; in particular, a characterization of the finite p-solvable groups having a character of this type (joint work with E. Pacifici and L. Sanus).
Title: Counting characters in blocks
Let G be a finite group, let p be a prime number and let B be a p-block of G with defect group D. Studying the structure of D by means of the knowledge of some aspects of B is a main area in character theory of finite groups. Let k(B) be the number of irreducible characters in the p-block B. It is well-known that k(B)=1 if, and only if, D is trivial. It is also true that k(B)=2 if, and only if, |D|=2. For blocks B with k(B)=3 it is conjectured that |D|=3.
In this talk we restrict our attention to the principal p-block of G, B_0(G), that is, the p-block containing the trivial character of G. In this case, by work of Belonogov, Koshitani and Sakurai we know the structure of D when k(B_0(G))=3 or 4. In this work, we go one step further and analyze the structure of D when k(B_0(G))=5.
This is a joint work with Mandi Schaeffer Fry and Carolina Vallejo.
Title: Word problems for finite nilpotent groups
We consider word maps on finite nilpotent groups and count the sizes of the fibres for elements in the image. We consider Amit’s conjecture and its generalisation, which say that these fibres should have size at least |G^(k−1)| where the word is on k variables. This is joint work with Ainhoa Iñiguez and Anitha Thillaisundaram.
Title: An Arad and Fisman's theorem on products of conjugacy classes revisited
There are several results about non-simplicity, solvability and normal structure of finite groups related to the product of conjugacy classes. In this framework it is well-known the Arad-Herzog's conjecture which asserts that a finite group having two classes such that its product is again a conjugacy class is not a non-abelian simple group. Now, we fix our focus on the case when the product of two classes is the union of these classes or one of these classes and the inverse of the other. These problems were originally studied by Arad and Fisman to obtain the non-simplicity of the group. They used elementary methods to prove this, however, recent results allow us to revisit these theorems and supply solvability and more structural properties within the group.
Joint work with Antonio Beltrán and María José Felipe
Title: On Huppert’s ρ-σ conjecture
The set of the degrees of the irreducible complex characters of a finite group G has been an object of considerable interest since the second part of the 20th century, and the study of the arithmetical structure of this set is a particularly intriguing aspect of Character Theory of finite groups. A remarkable question in this research area was posed by B. Huppert in the 80’s: is it true that at least one of the character degrees is divisible by a ”large” portion of the entire set of primes that appear as divisors of some character degree? More precisely, denoting by ρ(G) the set of primes that divide some character degree, and by σ(G) the largest number of primes that divide a single character degree, Huppert’s ρ-σ conjecture predicts that |ρ(G)| ≤ 3σ(G) holds for every finite group G, and that |ρ(G)| ≤ 2σ(G) if G is solvable. In this talk we will discuss some recent developments in the study of Huppert’s conjecture, obtained in a joint work with Z. Akhlaghi and S. Dolfi.
Title: On the order of products of elements in finite groups
It was proved by B. Baumslag and J. Wiegold that a finite group G is nilpotent if and only if o(x)o(y)=o(xy) for every pair of elements x,y of coprime order. In this talk, we will present several theorems that generalize this result.
Title: On a Conjecture of Malle and Navarro
Let G be a finite group and let P be a Sylow subgroup of G. In 2012 Malle and Navarro conjectured that P is normal in G if and only if the permutation character associated to the natural action of G on the cosets of P has some specific structural properties. In recent joint work with Law, Long and Vallejo we prove this conjecture. In this talk we will explain the main ideas involved in the proof. In particular we will discuss the importance of studying Sylow Branching Coefficients in this context.
Title: Hall-like theorems in products of π-decomposable groups
We discuss in this talk some Hall-like results for a finite group G = AB which is the product of two π-decomposable subgroups A = Aπ × Aπ′ and B = Bπ × Bπ′ , being π a set of odd primes. More concretely, we show that such a group G has a unique conjugacy class of Hall π-subgroups, and any π-subgroup is contained in a Hall π-subgroup (i.e. G satisfies property Dπ).
(Joint work with Lev S. Kazarin and M. Dolores Pérez-Ramos.)